A well-known phenomenon of human vision is that humans have different sensitivities to different colors. The sensors or receptors in the human eye are not equally sensitive to all wavelengths of light, and different receptors are more sensitive than others during periods of low light levels versus periods of relatively higher light levels. These receptor behaviors commonly are referred to as “scotopic” response (low light conditions), and “photopic” response (high light conditions). In the relevant literature, the scotopic response of human vision as a function of wavelength λ often is denoted as V′(λ) whereas the photopic response often is denoted as V(λ); both of these functions represent a normalized response of human vision to different wavelengths λ of light over the visible spectrum (i.e., wavelengths from approximately 400 nanometers to 700 nanometers). For purposes of the present disclosure, human vision is discussed primarily in terms of lighting conditions that give rise to the photopic response, which is maximum for light having a wavelength of approximately 555 nanometers.
A visual stimulus corresponding to a perceivable color can be described in terms of the energy emission of a light source that gives rise to the visual stimulus. A “spectral power distribution” (SPD) of the energy emission from a light source often is expressed as a function of wavelength λ, and provides an indication of an amount of radiant power per small constant-width wavelength interval that is present in the energy emission throughout the visible spectrum. The SPD of energy emission from a light source may be measured via spectroradiometer, spectrophotometer or other suitable instrument. A given visual stimulus may be thought of generally in terms of its overall perceived strength and color, both of which relate to its SPD.
One measure of describing the perceived strength of a visual stimulus, based on the energy emitted from a light source that gives rise to the visual stimulus, is referred to as “luminous intensity,” for which the unit of “candela” is defined. Specifically, the unit of candela is defined such that a monochromatic light source having a wavelength of 555 nanometers (to which the human eye is most sensitive) radiating 1/683 Watts of power in one steradian has a luminous intensity of 1 candela (a steradian is the cone of light spreading out from the source that would illuminate one square meter of the inner surface of a sphere of 1 meter radius around the source). The luminous intensity of a light source in candelas therefore represents a particular direction of light emission (i.e., a light source can be emitting with a luminous intensity of one candela in each of multiple directions, or one candela in merely one relatively narrow beam in a given direction).
From the definition above, it may be appreciated that the luminous intensity of a light source is independent of the distance at which the light emission ultimately is observed and, hence, the apparent size of the source to an observer. Accordingly, luminous intensity in candelas itself is not necessarily representative of the perceived strength of the visual stimulus. For example, if a source appears very small at a given distance (e.g., a tiny quartz halogen bulb), the perceived strength of energy emission from the source is relatively more intense as compared to a source that appears somewhat larger at the same distance (e.g., a candle), even if both sources have a luminous intensity of 1 candela in the direction of observation. In view of the foregoing, a measure of the perceived strength of a visual stimulus, that takes into consideration the apparent area of a source from which light is emitted in a given direction, is referred to as “luminance,” having units of candelas per square meter (cd/m2). The human eye can detect luminances from as little as one millionth of a cd/m2 up to approximately one million cd/m2 before damage to the eye may occur.
The luminance of a visual stimulus also takes into account the photopic (or scotopic) response of human vision. Recall from the definition of candela above that radiant power is given in terms of a reference wavelength of 555 nanometers. Accordingly, to account for the response of human vision to wavelengths other than 555 nanometers, the luminance of the stimulus (assuming photopic conditions) typically is determined by applying the photopic response V(λ) to the spectral power distribution (SPD) of the light source giving rise to the stimulus. For example, the luminance L of a given visual stimulus under photopic conditions may be given by:L=K(P1V1+P2V2+P3V3+ . . . ),  (1)where P1, P2, P3, etc., are points on the SPD indicating the amount of power per small constant-width wavelength interval throughout the visible spectrum, V1, V2, and V3, etc., are the values of the V(λ) function at the central wavelength of each interval, and K is a constant. If K is set to a value of 683 and P is the radiance in watts per steradian per square meter, then L represents luminance in units of candelas per square meter (cd/m2).
The “chromaticity” of a given visual stimulus refers generally to the perceived color of the stimulus. A “spectral” color is often considered as a perceived color that can be correlated with a specific wavelength of light. The perception of a visual stimulus having multiple wavelengths, however, generally is more complicated; for example, in human vision it is found that many different combinations of light wavelengths can produce the same perception of color.
Chromaticity is sometimes described in terms of two properties, namely, “hue” and “saturation.” Hue generally refers to the overall category of perceivable color of the stimulus (e.g., purple, blue, green, yellow, orange, red), whereas saturation generally refers to the degree of white which is mixed with a perceivable color. For example, pink may be thought of as having the same hue as red, but being less saturated. Stated differently, a fully saturated hue is one with no mixture of white. Accordingly, a “spectral hue” (consisting of only one wavelength, e.g., spectral red or spectral blue) by definition is fully saturated. However, one can have a fully saturated hue without having a spectral hue (consider a fully saturated magenta, which is a combination of two spectral hues, i.e., red and blue).
A “color model” that describes a given visual stimulus may be defined in terms based on, or related to, luminance (perceived strength or brightness) and chromaticity (hue and saturation). Color models (sometimes referred to alternatively as color systems or color spaces) can be described in a variety of manners to provide a construct for categorizing visual stimuli; some examples of conventional color models employed in the relevant arts include the RGB (red, green, blue) model, the CMY (cyan, magenta, yellow) model, the HSI (hue, saturation, intensity) model, the YIQ (luminance, in-phase, quadrature) model, the Munsell system, the Natural Color System (NCS), the DIN system, the Coloroid System, the Optical Society of America (OSA) system, the Hunter Lab system, the Ostwald system, and various CIE coordinate systems in two and three dimensions (e.g., CIE x,y; CIE u′,v′; CIELUV, CIELAB). For purposes of illustrating an exemplary color system, the CIE x,y coordinate system is discussed in detail below. It should be appreciated, however, that the concepts disclosed herein generally are applicable to any of a variety of color models, spaces, or systems.
One example of a commonly used model for expressing color is illustrated by the CIE chromaticity diagram shown in FIG. 1, and is based on the CIE color system. In one implementation, the CIE system characterizes a given visual stimulus by a luminance parameter Y and two chromaticity coordinates x and y that specify a particular point on the chromaticity diagram shown in FIG. 1. The CIE system parameters Y, x and y are based on the SPD of the stimulus, and also take into consideration various color sensitivity functions which correlate generally with the response of the human eye.
More specifically, colors perceived during photopic response essentially are a function of three variables, corresponding generally to the three different types of cone receptors in the human eye. Hence, the evaluation of color from SPD may employ three different spectral weighting functions, each generally corresponding to one of the three different types of cone receptors. These three functions are referred to commonly as “color matching functions,” and in the CIE systems these color matching functions typically are denoted as x(λ), y(λ), z(λ). Each of the color matching functions x(λ), y(λ), z(λ) may be applied individually to the SPD of a visual stimulus in question, in a manner similar to that discussed above in Eq. (1) above (in which the respective components V1, V2, V3 . . . of V(λ) are substituted by corresponding components of a given color matching function), to generate three corresponding CIE “primaries” or “tristimulus values,” commonly denoted as X, Y, and Z.
As mentioned above, the tristimulus value Y is taken to represent luminance in the CIE system and hence is commonly referred to as the luminance parameter (the color matching function y(λ) is intentionally defined to match the photopic response function V(λ), such that the CIE tristimulus value Y=L, pursuant to Eq. (1) above). Although the value Y correlates with luminance, the CIE tristimulus values X and Z do not substantially correlate with any perceivable attributes of the stimulus. However, in the CIE system, important color attributes are related to the relative magnitudes of the tristimulus values, which are transformed into “chromaticity coordinates” x, y, and z based on normalization of the tristimulus values as follows:x=X/(X+Y+Z)y=Y/(X+Y+Z)z=Z/(X+Y+Z)Based on the normalization above, clearly x+y+z=1, so that only two of the chromaticity coordinates are actually required to specify the results of mapping an SPD to the CIE system.
In the CIE chromaticity diagram shown in FIG. 1, the chromaticity coordinate x is plotted along the horizontal axis, while the chromaticity coordinate y is plotted along the vertical axis. The chromaticity coordinates x and y depend only on hue and saturation, and are independent of the amount of luminous energy in the stimulus; stated differently, perceived colors with the same chromaticity, but different luminance, all map to the same point x,y on the CIE chromaticity diagram. The curved line 50 in the diagram of FIG. 1 serving as the upper perimeter of the enclosed area indicates all of the spectral colors (pure wavelengths) and is often referred to as the “spectral locus” (the wavelengths along the curve are indicated in nanometers). Again, the colors falling on the line 50 are by definition fully saturated colors. The straight line 52 at the bottom of the enclosed area in the diagram, connecting the blue (approximately 420 nanometers) and red (approximately 700 nanometers) ends, is referred to as the “purple boundary” or the “line of purples.” This line represents colors that cannot be produced by any single wavelength of light; however, a point along the purple boundary nonetheless may be considered to represent a fully saturated color.
In FIG. 1, an “achromatic point” E is indicated at the coordinates x=y=1/3, representing full spectrum white. Hence, colors generally are deemed to become less saturated as one moves from the boundaries of the enclosed area toward the point E. FIG. 2 provides another illustration of the chromaticity diagram shown in FIG. 1, in which approximate color regions are indicated for general reference, including a region around the achromatic point E corresponding to generally perceived white light.
White light often is discussed in terms of “color temperature” rather than “color;” the term “color temperature” essentially refers to a particular subtle color content or shade (e.g., reddish, bluish) of white light. The color temperature of a given white light visual stimulus conventionally is characterized according to the temperature in degrees Kelvin (K) of a black body radiator that radiates essentially the same spectrum as the white light visual stimulus in question. Black body radiator color temperatures fall within a range of from approximately 700 degrees K (generally considered the first visible to the human eye) to over 10,000 degrees K; white light typically is perceived at color temperatures above 1500-2000 degrees K. Lower color temperatures generally indicate white light having a more significant red component or a “warmer feel,” while higher color temperatures generally indicate white light having a more significant blue component or a “cooler feel.”
FIG. 3 shows a lower portion of the chromaticity diagram of FIG. 2, onto which is mapped a “white light/black body curve” 54, illustrating representative CIE coordinates of a black body radiator and the corresponding color temperatures. As can be seen in FIG. 3, a significant portion of the white light/black body curve 54 (from about 2800 degrees K to well above 10,000 degrees K) falls within the region of the CIE diagram generally identified as corresponding to white light (the achromatic point E corresponds approximately to a color temperature of 5500 degrees K). As discussed above, color temperatures below about 2800 degrees K fall into regions of the CIE diagram that typically are associated with “warmer” white light (i.e., moving from yellow to orange to red).
One anomaly of human visual perception is that different colors or color temperatures (i.e., having different CIE chromaticity coordinates x and y) having a same luminance (i.e., a same CIE luminance parameter Y) actually may be perceived to have different brightnesses, even if perceived under the same photopic viewing conditions. This phenomenon is referred to in the relevant literature as the “Helmholtz-Kohlrausch” effect (hereinafter referred to as the HK effect). A variety of efforts have been made to model the HK effect (e.g., based on empirical data), and some exemplary discussions may be found in Nakano et al., “A Simple Formula to Calculate Brightness Equivalent Luminance,” CIE No. 133, CIE 24th Session, Warsaw, V.1, Part 1, pages 33-37, 1999; Natayani et al., “Perceived Lightness of Chromatic Object Color Including Highly Saturated Colors,” Color Res. Appl., 1, pages 127-141, 1992; Hunt, RWG, “Revised Colour-appearance model for related and unrelated colors,” Color Research Appl., 16, pages 146-165, and Natayani, Y., “A Colorimetric Explanation of the Helmholtz-Kohlrausch Effect,” Color Research Appl., Vol. 23, No. 6, 1998, each of which is incorporated herein by reference.
In general, according to the HK effect, saturated colors are perceived to be brighter than less saturated colors even when equal in luminance. Thus, if a white light and a saturated red light of the same luminance are compared side by side under the same viewing conditions, the red light looks brighter than the white to most observers. Similarly, if a white light and a saturated blue-green light of the same luminance are compared side by side, the blue-green light looks brighter than the white.
If, however, the saturated red and blue-green lights above are then added together and compared with the additive mixture of the two white lights above, the respective perceived brightnesses of the two mixtures are now similar; in this situation, the luminance of both mixtures is the same, and the perceived brightness of the mixtures also is the same. This arises because the mixture of the saturated red and blue-green light results in a whitish color, and the additional perceived brightness associated with the individual saturated colors has disappeared in the mixture. In view of the foregoing, while the luminance for different colors is additive, the perceived brightnesses of two different colors may not be additive.
An empirical formula has been developed (Kaiser, P. K., CIE Journal 5, 57 (1986)) that makes it possible to identify color stimuli which, on average, may be expected to be perceived as equally bright. First, a factor F is evaluated from the CIE chromaticity coordinates x and y corresponding to a given stimulus as follows:F=0.256−0.184y−2.527xy+4.656x3y+4.657xy4.  (2)Then, if two stimuli have respective luminances Y1 and Y2, and factors F1 and F2, the two stimuli are perceived with equal brightness if:log(Y1)+F1=log(Y2)+F2.  (3)If the left and right sides of Eq. (3) above are not equal, then whichever is greater indicates the stimulus having the greater perceived brightness. Similarly, it may be appreciated from Eq. (3) that, given equal luminance values Y1 and Y2 for two different stimuli, they will appear equally as bright to an observer if F1 equals F2.
FIG. 4 illustrates the CIE chromaticity diagram of FIG. 1, on which loci or “contours” 70A, 70B, 70C, etc., of equal values of F are shown based on Eq. (2) above. Again, two different colors falling into the same loci or contour appear equally as bright at the same luminance; hence, each contour indicated in FIG. 4 may be conceptually thought of as an “isobrightness” contour. The collection of isobrightness contours establishes the variation in perceived brightness across all chromaticity coordinates. The numbered values in FIG. 4 are given in terms of 10F for each contour. It may be observed by comparing FIGS. 2 and 4 that the nadir 70 of the contours (minimum value of 10F=0.836) occurs in a generally yellowish region of the CIE chromaticity diagram. From FIG. 4, it also may be appreciated that, pursuant to the HK effect, 10F generally tends to increase with increased saturation (i.e., as one moves from the yellowish region around the nadir 70 toward the spectral locus 50 or the purple boundary 52 of the diagram), especially in the direction of saturated reds, greens, and blues (refer again to FIG. 2).
Considering both sides of Eq. (3) as base-10 exponents, and re-writing Eq. (3) in terms of the values 10F, provides the relationships:
                                                        Y              1                        ⁢                          10                              F                1                                              =                                    Y              2                        ⁢                          10                              F                2                                                    ⁢                                  ⁢                              Y            2                    =                                                    10                                  F                  1                                                            10                                  F                  2                                                      ⁢                                          Y                1                            .                                                          (        4        )            The relationships in Eq. (4) illustrate that the numeric values assigned to the contours in FIG. 4 provide factors by which the luminance of a first stimulus having a chromaticity lying in one of the contours may be adjusted (increased or decreased) relative to the luminance of a second stimulus having a chromaticity lying in a different contour, so that both stimuli appear to have the same brightness when seen under the same viewing conditions.
Accordingly, the collection of isobrightness contours given by Eq. (2) and the corresponding relationships in Eq. (4) establish the variation in perceived brightness across all chromaticity coordinates. For example, consider a first stimulus having a luminance Y1 and chromaticity coordinates that fall in the contour 70B corresponding to 10F=1, and a second stimulus having a luminance Y2 and chromaticity coordinates that fall in the contour 70G corresponding to 10F=1.5. For these two stimuli to be perceived as having the same brightness, according to Eq. (4) the luminance Y2 needs to be (1/1.5) or 0.667Y1.